Ant highway with energy loss

Calculus Level 3

An ant starts at the center of the image. The ant only has enough energy to walk for 10 meters (1 meter = 1 grid) on one of the highways (directly north, east, south, and west) before stopping.

However, if it goes off the highway, more energy is needed for the same distance. That is, the further the ant deviates, the greater the energy consumption, as represented by the factor \(1+0.4d\), where \(d\) is the distance in meters to the nearest highway. For example, if the ant is 1 meter away from the nearest highway, it uses 1.4 times the energy on the highway to walk 1 meter.

What is the area (in square meters) of the set of points the ant can reach before getting tired and stopping?

The above is one possible path (in red). The ant walks 5 meters north and then curves off towards the right.

The above is one possible path (in red). The ant walks 5 meters north and then curves off towards the right.

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